We are given that AD is perpendicular to CE. Also, we know that CE and AD are the medians of â–³ ABC. This means that E and D are the midpoints of AB and CB respectively. Let's mark these pieces of information on the given diagram.
As we can see, â–³ AOC is a right triangle. Thus, to find AC we can use the Pythagorean Theorem. Although, first we need to find CO and AO. Let's do this!
Finding CO
We are going to use the fact that CE and AD are the medians of â–³ ABC. A point of intersection of triangle medians, which in our case is O, is called a centroid. Let's recall what the Centroid Theorem states.
The medians of a triangle inersect
at a point called the centroid that is
two thirds of the distance from each vertex
to the midpoint of the opposite side.
According to this theorem, CO is two thirds of the distance from C to the midpoint E.
CO=2/3CE
It is given that segment CE measures 9. By substituting this value into the above equation we can calculate CO.
We can see that segment CE consists of CO and OE. By the Segment Addition Postulate, its measure is the sum of measures of CO and OE.
CE= CO+OE
It is given that CE measures 9 and we have found that CO measures 6. By substituting these values into this equation, we can find the measure of OE.
We also know that AB measures 10. Because CE is a median of AB, segments AE and EB are congruent and have the same measure. Dividing 10 by 2, we get that each of them measures 5.
Let's now consider the triangle △ AOE. It is a right triangle, as CE and AD are perpendicular and form a right angle ∠AOE. Hence, we can apply to it the Pythagorean Theorem.
AO^2+OE^2=AE^2
We know the values of OE and AE, so we can substitute OE with 3 and AE with 5.
AO^2+3^2=5^2
Let's solve this equation and find AO.
Now that we know the measures of CO and AO, we can find AC.
Let's use the Pythagorean Theorem, which applied to â–³ AOC has the following form.
AO^2+ CO^2=AC^2
If we substitute AO with 4 and CO with 6, we will get the equation where the only unknown is AC.
4^2+ 6^2=AC^2
Let's solve it!
